New superpermutations at n=6, L=872, with repeated permutations.

[Tomaz Kristan]

One repeated in four files and two in two files. Not a Hamiltonian way of doing things.

Claude Opus 5 was quite helpful and I asked him if he want to be mentioned and he said he would.

[Shea Street]

Thanks again for posting more finds of duplicate permutations. Since I was already rechecking the one you posted for n=7 5906 I went ahead and took a look at these. I am again using zero indexing.

-872_r1_g2_1 has a duplicate: 235416 at 499 and 505

-872_r1_g2_2 has a duplicate: 654132 at 564 and 570

-872_r1_g2_3 has a duplicate: 235416 at 441 and 447

-872_r1_g2_4 has a duplicate: 654132 at 506 and 512

-872_r2_g3_1 has two duplicates:  235416 at 499 and 505 & 654132 at 564 and 570

-872_r2_g3_2 has two duplicates:  235416 at 441 and 447 & 654132 at 506 and 512

Thanks again for sharing your interesting finds. I know I am not telling you anything you don't already know but I like to examine things again with different tools/code as well as share to the community a second pair of eyes.

[Tomaz Kristan]

Thanks, Sheam!

[Miles Gould]

Oh, that's interesting! I'd thought that the repeated permutations you'd found in n=7 superpermutations meant that length 5906 probably wasn't optimal, but since it looks like 872 is optimal for n=6 then maybe duplicate permutations are not disqualifying? How many of the known length-872 superpermutations have duplicates?

Thanks,

Miles

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[Shea Street]

For myself it wasn't until recent months that I even dropped that permutations must only show up unique in a viable (near or exact) minimum. There is nothing prohibitive for a superpermutation to have duplication I think it's just been held that it is inefficient for minimum as an instinctual thing. Even if they are permissible they may not shown up in known records due to their construction lineages. That does not say they are not produced as byproducts, they just might have never been the focus and thus not inspected for since most might only check for a superpermutation overall length and that it contains all permutations and not necessarily counts of each beyond existing. I been thinking on data mining repo records at some point to recheck existing to see if there is anything to glean.

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[Tomaz Kristan]

Miles:

>  How many of the known length-872 superpermutations have duplicates?

Of 50018, a few days ago, only those 6 of mine. We are talking about L=872. Two of those 6 have double doubles inside. Means not only 1 permutation duplicated, but two are both duplicated. And the both same copies are always next to each other!